Link Formation in Cooperative Situations by Bhaskar Dutta Indian Statistical Institute 7SJS Sansanwal Marg, New Delhi-110016, India e-mail
[email protected] Anne van den Nouweland Department of Econometrics, and Center for Economic Research Tilburg University, PO Box 90153, 5000 LE Tilburg, The Netherlands e-mail
[email protected] Stef Tijs Department of Econometrics, and Center for Economic Research Tilburg University, PO Box 90153, 5000 LE Tilburg, The Netherlands e-mail
[email protected]
February, 1995
Abstract : In this paper we study the endogenous formation of cooperation structures or communication graphs between players in a superadditive TU game. For each cooperation structure that is formed, the payo s to the players are determined by an exogenously given solution. We model the process of cooperation structure formation as a game in strategic form. It is shown that several equilibrium re nements predict the formation of the complete cooperation structure or some structure which is payo -equivalent to the complete structure. These results are obtained for a large class of solutions for cooperative games with cooperation structures. A by-product of our analysis is a characterization of the class of weighted Myerson values.
1
1
Introduction
Usually, cooperative game theory is concerned with predicting how rational players will distribute the gains that are obtained through cooperation. The standard approach in the literature is to represent the underlying situation as a game in characteristic function form. A solution concept speci es the distribution of payo s for each game. This formulation (usually) either implicitly assumes that the grand coalition will form, or speci es an exogenous coalition structure. However, the distribution of payo s will depend on the structure of coalitions which form, since this will typically determine the total amount that is available for distribution. Moreover, the eventual coalitional structure itself will usually be in uenced by what players expect to get in di erent coalitions. Hence, the ideal approach is one in which the coalition structure as well as the distribution of payo s are determined simultaneously. It is natural in this context to adopt the so-called Nash program, and try and support the prediction of any endogenous theory of coalition formation and associated solution concept as a noncooperative equilibrium outcome of a `larger' game in which the negotiation process is embedded. Since the seminal work of Rubinstein (1982), there have been a number of papers in this tradition. Of particular relevance for present purposes are Binmore (1985), Chatterjee et al. (1993), Gul (1989), Perry and Reny (1994) and Selten (1981), where the negotiation process associated with characteristic function games is explicitly modelled. While Gul (1989) and Perry and Reny (1994) derived negotiation processes leading up to speci c solution concepts (the Shapley value and the core respectively), Chatterjee et al. (1993) formulated a generalization of the Rubinstein alternating o ers model to represent the negotiation process. Amongst other results, they also showed that the grand coalition need not always form even in strictly superadditive games. Of course, all these papers
2 provide an endogenous theory of coalition formation as well as a prediction about the distribution of payo s.1 There are two points of departure from this literature in the current paper. First, we focus attention on Myerson's (1977) cooperation structures2 , rather than coalition structures. A cooperation structure is a graph whose vertices are identi ed with the players. A link between two players means that these players can carry on meaningful direct negotiations with each other. Notice that a coalition structure is a special kind of cooperation structure where two members i and j are linked if and only if they are in the same coalition.3 Second, following Aumann and Myerson (1988), we model situations in which the eventual distribution of payo s is determined in two distinct stages or periods. The rst period is devoted to link formation only. During this period, the players cannot enter into binding agreements of any kind, either on the nature of the link formation, or on the subsequent division of payo s. In the second period, no new links can be formed, but players negotiate over the division of the payo , given the cooperation structure which has formed in the rst stage. The goal of this paper is to analyse the endogenous formation of cooperation structures in this setting. In order to do this, we assume that in the rst stage of the above process, agents' decisions on whether or not to form a link with other agents can be represented as a game in strategic form.4 In the link 1 See
also Sengupta and Sengupta (1994), who determine the coalition structure and
distribution of payo s simultaneously, although not in the tradition of the Nash program. 2 See van den Nouweland (1993) for a survey of recent research on games with coooperation structures. 3 Aumann and Myerson (1988) give examples of negotiation situations which can be modelled by cooperation structures, but not by coalition structures. 4 This game was originally introduced by Myerson (1991) (p. 448). See also Hart and Kurz (1983), who discuss a similar strategic-form game in the context of the endogenous formation of coalition structures. In contrast, Aumann and Myerson (1988) model the process of link
3 formation game, each player announces a set of players with whom he or she wants to form a link. A link is formed between i and j if both players want the link. Given the announcements of the n players, this speci cation gives the cooperation structure. Suppose there is a rule or solution which determines a distribution of payo s for each cooperation structure. This, then, also gives the payo function of the strategic form game. Since this is a well-de ned strategic form game, we can use any noncooperative equilibrium concept to analyse the game. Suppose now that the rule which determines payo s for each cooperation structure has the property that no agent wants to unilaterally break a link with any player. Since no player wants to break a link, and it needs the consent of two players to form an additional link, any cooperation structure can be sustained as a Nash equilibrium. We, therefore, use re nements of the Nash equilibrium concept. In particular, we employ undominated Nash equilibrium, coalition-proof equilibrium, and the argmax set of weighted potential games.5 Our principal conclusion is that for a wide class of solutions, these equilibrium re nements all lead to the formation of the full cooperation structure or cooperation structures which are payo -equivalent to this structure. An important by-product of our analysis is a characterization of weighted Myerson values which are a generalization of weighted Shapley values to games with cooperation structures. We show that weighted Myerson values are the only solution concepts for games with cooperation structurs which satisfy an eciency requirement and which generate linking games that are weighted potential games. The plan of this paper is as follows. In section 2 we provide some basic de nitions, including those of cooperation structures and solutions for games with cooperation structures. Some `reasonable' properties on such solutions formation as a game of perfect information. We discuss this issue in more detail in section 3. 5 The
latter is de ned in section 5.
4 are introduced, and some implications are derived. Section 3 contains a discussion of di erent ways of modelling the process of link formation. Endogenous cooperation structures corresponding to undominated Nash equilibrium and coalition-proof Nash equilibrium are determined in section 4. Section 5 contains the characterization of weighted Myerson values, and also shows that the argmax set of the weighted potential corresponds to the full cooperation structure and payo -equivalent structures. We conclude in section 6.
2
Cooperation Structures and Solutions
Let (N; v ) be a T U coalitional game, where N = f1; 2; : : : ; ng denotes the nite player set and v is a real-valued function on the family 2N of all subsets of N
with v(;) = 0. Throughout this paper, we will assume that v is superadditive6 . A cooperation structure is a graph g = (N; L) where N is the set of vertices, and L is the edge set. An edge will also be called a link, and denoted by l; l etc. For any S
0
N , we say that players i; j 2 S are connected in S if there
exists a path from i to j that uses only vertices in S . The relation `connected in N ' is an equivalence relation on N . The equivalence classes of this relation are the connected components of the graph g . We follow Aumann and Myerson (1988) in interpreting a link between two players as meaning that these players can carry on meaningful direct negotiations with each other. The negotiation to form links takes place in a preliminary period when \for one reason or another, one cannot enter into binding agreements of any kind (such as those relating to subsequent divisions of the payo ..)".7
A solution is a mapping which assigns an element in IRn to each T U
6v
is superadditive if for all S; T
7 Aumann
2 2N with \ S
T
= ;; v(S ) + v (T ) v (S [ T ).
and Myerson (1988), page 187. See also Myerson (1977).
5 game (N; v) and cooperation structure g = (N; L). Since there will be no ambiguity about the underlying game (N; v ), we will simply write (L); (L ), 0
etc., instead of writing (N; v; L); (N; v; L ), etc. 0
A solution can for example be generated for any graph g by applying the usual or familiar cooperative solution concepts to the `graph-restricted game' (N; vg ). This game is de ned as follows. Let S ng denote the partition of S into subsets of players that are connected in S by g . That is, S ng = ffi j j and i are connected in S by g
Now, de ne vg : 2N
! IR by v g (S ) =
X T Sg 2
g j j 2 Sg
v(T )
(1)
(2)
n
For instance, for any g = (N; L), the Shapley value of the associated game (N; v g ) is a solution for (N; v; L), and has come to be called the Myerson value.8 Similarly, weighted Myerson values of (N; v; L) are the weighted Shapley values of (N; vg ). A class of solutions which will play a prominent role in this paper is the class satisfying the following `reasonable' properties on a solution below.
Component eciency (CE) : For all cooperation structures (N; L) and all S
2 2N , if S is a connected component of (N; L), then
X i S
i (L) = v(S ).
2
Weak link symmetry (WLS) : For all i; j 2 N , and all cooperation structures (N; L), if i (L [ fi; j g) > i (L), then j (L [ fi; j g) > j (L).
Improvement property (IP) : For all i; j 2 N and all cooperation structures (N; L), if for some k
2 N nfi; j g;
i (L) or j (L [ fi; j g) > j (L).
k (L [ fi; j g) > k (L), then i (L [ fi; j g) >
8 Myerson (1977) contains a characterization of the Myerson value. See also Jackson and Wolinsky (1994).
6 These properties all have very simple interpretations. Component eciency, which was originally used by Myerson (1977), states that the players in a connected component S split the value v (S ) amongst themselves. The second property is a very weak form of symmetry. It says that if a new link between players i and j makes i
strictly
better o , then it must also strictly improve
the payo of player j . Finally, the improvement property states that if a new link between players i and j strictly improves the payo of any other player k , then the payo of either i or j must also strictly improve. The class of weighted Myerson values satis es all the properties listed above. There are also others. For instance, if (N; v) is a tarian solution
convex
game, then the egali-
of Dutta and Ray (1989) corresponding to the associated game
(N; v g ) also satis es these properties.
The three properties together imply an interesting fourth property. This is the content of the next lemma.
Lemma 1 i; j
2 N;
Let
be any solution satisfying CE, WLS and IP. Then, for al l
and all cooperation structures
(N; L),
i (L [ fi; j g) i (L):
Proof :
Suppose for some i; j
2
N and (N; L), i (L) > i (L
(3)
[ fi; j g).
j (L [ fi; j g). But then, since v is superadditive, and satis es CE, there must exist k 62 fi; j g such that
k (L) < k (L [fi; j g). This shows that violates IP since i (L) > i (L [fi; j g) and j (L) j (L [ fi; j g). Then, by WLS, we must also have j (L)
Remark 1: We will denote the property incorporated in equation (3) by Link Monotonicity
. Note that Link Monotonicity is an appealing property in
its own right. It says that a player i should not be worse-o as a result of
7 forming a new link with some player j .9
Remark 2: It is easy to construct examples to show that the Component Eciency, Weak Link Symmetry, and Improvement properties are independent. Another consequence of these three properties is derived in the next lemma.
We show that if the formation of a link fi; j g a ects the payo of some other
player k , then it must also a ect the payo s of both players that formed the new link. This property will be used later on in the paper.
Lemma 2
Let
satisfy CE, WLS and IP. Then, for al l
cooperation structures then
i; j
2
N,
and al l
(N; L), if for some k 2 N nfi; j g, k (L [fi; j g) 6= k (L),
i (L [ fi; j g) 6= i (L)
and
j (L [ fi; j g) 6= j (L).
Proof: Suppose for some i; j 2 N and k 2 N nfi; j g, k (L [ fi; j g) 6= k (L). If
k (L [ fi; j g) > k (L), then from WLS and IP we must have i (L [ fi; j g) >
i (L) and j (L [ fi; j g) > j (L).
Suppose k (L [fi; j g) < k (L). From WLS, either i (L [fi; j g) > i (L) and
j (L [fi; j g) > j (L), or i (L [fi; j g) i (L) and j (L [fi; j g) j (L). But,
in the latter case, CE and superadditivity imply that there exists a l
62 fi; j g
such that l (L [ fi; j g) > l (L). This would violate IP, so it must hold that
i (L [fi; j g) > i (L) and j (L [fi; j g) > j (L). This establishes the lemma.
While the three properties are all appealing and are satis ed by a large class of solutions, there are other solutions outside this class that seem to be appealing. One such solution is de ned below.
For any i and L; let Li = ffi; j g j j 2 N; fi; j g 2 Lg, the set of links that are adjacent to i, and li = jLi j. Let Si (L) denote the connected component of 9 Note that the game is superadditive.
8 L containing i. Then, the Proportional
8> < Pj2Sli L lj v(Si(L)) P i
i (L) = > : v(fig) ( )
for all L and all i
2 N.
, denoted P , is given by
Links Solution
if jSi (L)j 2 if Si (L) = fig
(4)
The solution P captures the notion that the more
links a player has with other players, the better are his
relative
prospects in
the subsequent negotiations over the division of the payo . Notice that this makes sense only when the players are equally `powerful' in the game (N; v). Otherwise, a
big
player may get more than
smal l
players even if he has fewer
links. We leave it to the reader to check that P satis es CE and IP, but not WLS.
3
Modelling Negotiation Processes
In this section, we use the 3-person majority game to illustrate some of the issues involved in the endogenous formation of cooperation structures. In particular, we discuss the Aumann-Myerson extensive form approach in some detail. We point out that this approach, which involves a sequential formation of links may be appropriate in situations where the negotiation process is `public', and where for one reason or another, bilateral negotiations take place in some predetermined order. When these prerequisites are not met, it may be more appropriate to model the `negotiation game' as a game in strategic form. This is the approach adopted in this paper, and it is de ned more formally later on in this section. Aumann and Myerson (1988) use the following extensive form. First, an exogenous
rule
determines the sequential order in which pairs of players nego-
tiate to form a link. A link is formed if and only if both potential partners agree, and once formed, a link cannot be broken. Moreover, after the last pair
9 in the order has decided on whether or not to form a link, all the remaining pairs are given another opportunity to form a link. The process stops if all pairs that did not form a link yet have had a last opportunity to do so. At any point of time, the entire history of links formed or rejected is known to the players, so that it is a game of perfect information. Some cooperation graph g will form at the end of the process. The payo to player i is then i (vg ), that is, the Shapley value of player i in the game v g .
Since the game is nite and of perfect information, it has subgame perfect equilibria in pure strategies. The `prediction' of the model is that only cooperation structures associated with subgame perfect equilibria will actually form as a result of negotiations between players.
Consider, for instance, the TU game v on the player set f1; 2; 3g de ned by
8> < 1 if j S j 2 v (S ) = > : 0 otherwise
Suppose also that the rule speci es that the order of pairs is f1; 2g; f1; 3g; f2; 3g. Then, the Aumann-Myerson prediction is that only one pair will form a link. Suppose the link f1; 2g is formed. Notice that either of 1 and 2 gain by forming
an additional link with 3, provided the other player does not form a link with 3. Two further points need to be noted. First, if player i forms a link with 3, then it is in the interest of j (j = 6 i) to also link up with 3. Second, if
al l
links are
formed, then players 1 and 2 are worse-o compared to the graph in which they
alone form a link. Hence, the structure ff1; 2gg is sustained as an `equilibrium' by a pair of mutual threats of the kind : \If you form a link with 3, then so will I." Of course, this kind of threat makes sense only if i will come to know whether j has formed a link with 3. Moreover, i can acquire this information only if the negotiation process is
public
. If bilateral negotiations are conducted secretly,
10 then it may be in the interest of some pair to conceal the fact that they have formed a link until the process of bilateral negotiations has come to an end. It is also clear that if di erent pairs can carry out negotiations simultaneously and if links once formed cannot be broken, then the mutual threats referred to earlier cannot be carried out.10 Thus, there are many contexts where considerations other than threats may have an important in uence on the formation of links. For instance, suppose players 1 and 2 have already formed a link amongst themselves. Suppose also that neither player has as yet started negotiations with player 3. If 3 starts negotiations simultaneously with both 1 and 2, then 1 and 2 are in fact faced with a Prisoners' Dilemma situation. To see this, denote l and nl as the strategies of forming a link with 3 and not forming a link with 3 respectively. Then, the payo s to 1 and 2 are described by the following matrix (the rst entry in each box is 1's payo , while the second entry is 2's payo ). Player 2 l
Player 1
nl
l
( 13 ; 13 ) ( 23 ; 16 )
nl
( 16 ; 23 ) ( 12 ; 12 )
Note that l, that is forming a link with 3, is a dominant strategy for both players! Obviously, the complete graph may well form simply because players 1 and 2 cannot sign a binding agreement to abstain from forming a link with 3. In this paper, we model the negotiation process as a game in strategic form. The speci c strategic form game that we will construct was rst de ned by Myerson (1991), and has subsequently been used by Qin (1993). This model is described below. 10 Aumann and Myerson (1988) also stress the importance of perfect information in deriving
their results.
11 Let be a solution. Then, the linking game 0( ) associated with is given by the (n + 2)-tuple (N ; S1 ; : : : ; Sn ; f ) where for each i 2 N , Si is player i's strategy set with Si = 2N nfig , and the payo function is the mapping f : 5i2N Si ! IRn given by fi (s) = i (L(s))
(5)
for all s 2 5i2N Si , with L(s) = ffi; j g j j
2
si ; i 2 sj g
(6)
The interpretation of (5) and (6) is straightforward. A typical strategy of player i in 0( ) consists of the set of players with whom i wants to form a link. Then (6) states that a link between i and j is formed if and only if they both want to form this link. Thus, each strategy vector s gives rise to a unique cooperation structure L(s). Finally, the payo to player i associated with s is simply i (L(s))11 , the payo that associates with the cooperation structure L(s). We will let s = (s1; :::; sn ) denote the strategy vector such that si = N nfig for all i 2 N , while L = ffi; j g j i 2 N; j 2 N g = L(s) denotes the complete edge set on N . A cooperation structure L is essentially complete for if ). Hence, if L is essentially complete for , but L 6= L , then the
(L) = (L links which are not formed in L are inessential in the sense that their absence does not change the payo vector from that corresponding to L . Notice that the property of \essentially complete" is speci c to the solution - a cooperation structure L may be essentially complete for , but not for 0 . We now de ne some equilibrium concepts for any 0( ). These will be used in section 4 below. 11 We
again remind the reader that we have suppressed the underlying T U game (N ; v ) in
order to simplify the notation.
12 The rst equilibrium concept that we consider is the undominated Nash equilibrium. For any i 2 N; si dominates s0i i for all s0i 2 S0i ; fi (si ; s0i ) fi (s0i ; s0i ) with the inequality being strict for some s0i . Let Siu ( ) be the set
of undominated strategies for i in 0( ), and S u ( ) = 5i2N Siu ( ). A strategy tuple s is an undominated Nash equilibrium of 0( ) if s is a Nash equilibrium and, moreover, s 2 S u ( ). The second equilibrium concept that will be discussed is the CoalitionProof Nash Equilibrium. In order to de ne the concept of Coalition-Proof Nash Equilibrium of 0( ), we need some more notation. For any T N and s3T
2
ST := 5i2T Si , let 0( ; s3N nT ) denote the game induced on subgroup T by
the actions s3N nT . So,
0( ; s3N nT ) = hT; fSi gi2T ; f~ i where for all j 2 T , f~j : 5i2T Si ! IR is given by f~j ((si )i2T ) = fj ((si )i2T ; s3N nT ) for all (si )i2T 2 ST . The Coalition-Proof Nash Equilibrium is de ned inductively as follows: In a single player game, s3 2 S is a Coalition-Proof Nash Equilibrium (CPNE) of 0( ) i s3i maximizes fi (s) over S . Now, let 0( ) be a game with n players, where n > 1, and assume that Coalition-Proof Nash Equilibria have been de ned for games with less than n players. Then, a strategy tuple s3 2 SN := 3 5i2N Si is called self-enforcing if for all T 6= N; sT is a CPNE in the game
0( ; s3N nT ). A strategy tuple s3 2 SN is a CPNE of 0( ) if it is self-enforcing and, moreover, there does not exist another self-enforcing strategy vector s 2 SN such that fi (s) > fi (s3 ) for all i 2 N . Let CPNE ( ) denote the set of CPNE of 0( ):12 Notice that the notion of CPNE incorporates a kind of `farsighted' thought process on the part of players 12 See
Bernheim, Peleg and Whinston (1987) for discussion of Coalition-Proof Nash
Equilibrium.
13 since a coalition when contemplating a deviation takes into consideration the possibility of further deviations by subcoalitions.13
4
Equilibrium Cooperation Structures
In this section, we characterize the sets of equilibrium cooperation structures under the equilibrium concepts de ned in the previous section. Our principal objective is to show that the equilibrium concepts de ned in section 3 all lead to essentially complete cooperation structures for solutions satisfying the properties that are listed in section 2. Theorem 1 Let be a solution that satis es CE, WLS and IP. Then, s is
an undominated Nash equilibrium of 0( ). Moreover, if s is an undominated Nash equilibrium of 0( ), then L(s) is essentially complete for . Proof : First, we show that si is undominated for all i 2 N (in fact, we even
show that it is weakly dominant). So, choose i 2 N; si 2 Si and s0i 2 S0i arbitrarily. Let L = L(si ; s0i ) and L0 = L(si ; s0i ). Note that, since si si , L0 L. Also, if l 2 LnL0 , then i 2 l. So, from repeated application of link monotonicity (see lemma 1), fi ( si ; s0i ) = i (L) i (L0 ) = fi (si ; s0i )
(7)
Since si and s0i were chosen arbitrarily, this shows that si 2 Siu ( ). Further, putting s0i = s0i in (7), we also get that s is a Nash equilibrium of 0( ). So, we may conclude that s 2 S u ( ). 13 We
mention this because Aumann and Myerson (1988) state that they do not use the
`usual, myopic, here-and-now kind of equilibrium condition', but a `look ahead' one. Of course, farsightedness can be modelled in many di erent ways.
14 Now, we show that L(s) is essentially complete for an undominated Nash equilibrium s. Choose s = 6 s arbitrarily. Without loss of generality, let fi N
j
2
si = 6 si g = f1; 2; :::; K g. Construct a sequence fs0 ; s1 ; : : : ; sK g of strategy
tuples as follows. (i) s0 = s (ii) skk = sk for all k = 1; 2; :::; K:
(iii) skj = skj 01 for all k = 1; 2; :::; K , and all j 6= k .
Clearly, sK = s. Consider any sk 01 and sk . By construction, skj 01 = skj for all j 6= k , while skk = sk and skk01 = sk . So, using link monotonicity, we have fk (sk ) = k (L(sk )) k (L(sk01)) = fk (sk01 )
(8)
Suppose (8) holds with strict inequality. Then, we have demonstrated the existence of strategies s0k such that fk ( sk ; s0k ) > fk (sk ; s0k )
(9)
But, (7) and (9) together show that sk dominates sk . So, if s 2 S u ( ), then (8) must hold with equality. Then it follows from lemma 2 that the payo s to all players remain unchanged when going from sk01 to sk , so
(L(sk )) = (L(sk01 ))
(10)
Since this argument can be repeated for k = 1; 2; :::; K , we get (L(s0)) =
(L(s1 )) = 1 1 1 = (L( s)). Hence, if s 2 S u ( ), then L(s) is essentially complete. Theorem 2 Let be a solution satisfying CE, WLS and IP. Then s
CPNE ( ). Moreover, if s for .
2
2
CPNE ( ), then L(s) is essentially complete
15 Proof : In fact, we will prove a slightly generalized version of the theorem
and show that for each coalition T sT
2
N
and all sN nT
CPNE ( ; sN nT ) and that for all s3T
2 CPNE
2 SN nT
it holds that
( ; sN nT ) it holds that
f (s3T ; sN n T ) = f ( sT ; sN nT ). We will follow the de nition of Coalition-Proof
Nash Equilibrium and proceed by induction on the number of elements of T . Throughout the following, we will assume sN nT
2 SN nT to be arbitrary.
Let T = fig. Then by repeated application of Link Monotonicity we know that fi (si ; sN nfig ) fi (si ; sN nfig ) for all si 2 Si . From this it readily follows that si 2 CPNE ( ; sN nfig ). Now, suppose s3i 2 CPNE ( ; sN nfig). Then, since fi (s3i ; sN nfig )
fi ( si ; sN nfig ), it follows that fi (s3i ; sN nfig) = fi ( si ; sN nfig)
must hold. Now we use lemma 2 and see that f (s3i ; sN nfig ) = f (si ; sN nfig ).
Now, let jT j > 1 and assume that we already proved that for all R with jRj < jT j and all sN nR 2 SN nR it holds that sR 2 CPNE ( ; sN nR) and that for all s3R 2 CPNE ( ; sN nR ) it holds that f (s3R ; sN nR ) = f (sR ; sN nR ). Then
it readily follows from the rst part of the induction hypothesis that sR 2 CPNE ( ; sT nR ; sN nT ) for all R 6= T . This shows that sT is self-enforcing. Suppose s3T 2 ST is also self-enforcing, i.e. s3R 2 CPNE ( ; s3T nR ; sN nT )
3 for all R 6 T . We will start by showing that fi (sT ; sN nT ) fi (sT ; sN nT ) = for all i 2 T , which proves that sT 2 CPNE ( ; sN nT ). So, let i 2 T be xed for the moment. Then repeated application of Link Monotonicity implies that fi (sT ; sN nT ) fi (s3i ; sT ni ; sN nT ). Further, since s3T nfig 2
CPNE ( ; s3i ; sN nT ), it follows from the second part of the induction hypothesis that f (s3i ; sT nfig ; sN nT ) = f (s3T ; sN nT ). Combining the two last (in)equalities we nd that fi (sT ; sN nT ) fi (s3T ; sN nT ). Note that we will have completed the proof of the theorem if we
show that, in addition to fi (sT ; sN nT )
fi (s3T ; sN nT ) for all i
2
T , it
16 holds that either fi (sT ; sN nT ) > fi (s3T ; sN nT ) for all i 2 T (and, consequently, s3T 62 CPNE ( ; sN nT ) ) or fi (sT ; sN nT ) = fi (s3T ; sN nT ) for all CPNE ( ; sN nT ) ). So, suppose i 2 T is such that fi ( sT ; sN nT ) > fi (s3T ; sN nT ). Because s3T is self-enforcing, we know that
i
2
T (and s3T
s3T nfj g
2
2
CPNE ( ; s3j ; sN nT ) for each j
2
T , and it follows from the in-
duction hypothesis that f (s3T ; sN nT ) = f (s3j ; sT nj ; sN nT ) for each j
2
T.
Let j 2 T nfig be xed. Then we have just shown that fi (sT ; sN nT ) > fi (s3T ; sN nT ) = fi (s3j ; sT nj ; sN nT ). We know by repeated application of Link Monotonicity that fj (sT ; sN nT ) fj (s3j ; sT nj ; sN nT ). However, if this should hold with equality, fj (sT ; sN nT ) = fj (s3j ; sT nj ; sN nT ) then repeated application of lemma 2 would imply that f (sT ; sN nT ) = f (s3j ; sT nj ; sN nT ), which contradicts that fi (sT ; sN nT ) > fi (s3j ; sT nj ; sN nT ). Hence, we may conclude that
fj ( sT ; sN nT ) > fj (s3j ; sT nj ; sN nT ). Since fj (s3j ; sT nj ; sN nT ) = fj (s3T ; sN nT ), we
now know that fj (sT ; sN nT ) > fj (s3T ; sN nT ). This shows that either fi (sT ; sN nT ) > fi (s3T ; sN nT ) for all i fi ( sT ; sN nT ) = fi (s3T ; sN nT ) for all i 2 T .
2
T or
Remark 3: We have an example of a solution satisfying CE, WLS and
IP, for which CPNE ( ) 6= fs j L(s) is essentially completeg. In other words, there may be s which is not in CPNE ( ), though L(s) is essentially complete. We de ned the Proportional Links Solution P in section 2, and pointed out that it does not satisfy WLS. It also turns out that the conclusions of theorem 2 are no longer valid in the linking game 0( P ). While we do not have any general characterization results for 0( P ), we show below that complete structures will not necessarily be coalitionproof equilibria of 0( P ) in the special case of the 3-player majority game.14 14 v
is a majority game if a majority coalition has worth 1, and all other coalitions have
zero worth.
17 Proposition 1 Let N be a player set with jN j = 3, and let v be the majority
game on jN j. Then, s 2 CPNE ( P ) i L(s) = ffi; j gg, i.e., only one pair of agents forms a link. P
P
Proof : Suppose only i and j form a link according to s. Then, fi (s) =
fj (s) = 21 . Check that if i deviates and forms a link with k , then i's payo
remains at 21 . Also, clearly i and j together do not have any pro table deviation. Hence, s is coalitionproof. Now, suppose that N is a connected set according to s. There are two possibilities. P Case (i) : L(s) = L . In that case, fi = 31 for all i 2 N . Let i and j deviate and break links with k . Then, both i and j get a payo of 21 . Suppose i makes a further deviation. The only deviation which needs to be considered is if i re-establishes a link with k . Check that i's payo remains at 21 . So, in this case s cannot be a coalition proof equilibrium. Case (ii) : L(s) 6= L . Since N is a connected set in L(s), the only possibility is that there exist i and j such that both are connected to k , but not to each other. Then, both i and j have a payo of 41 . Let now i and j deviate, break links with k and form a link between each other. Then, their payo increases to 21 . Check that neither player has any further pro table deviation. Again, this shows that s is not coalitionproof.
5
Weighted Potential Games
Monderer and Shapley (1993) prove various properties of the class of potential games.15 Their results make potential games particularly interesting, and 15 Rosenthal
gic form.
(1973) was the rst to (implicitly) use potential functions for games in strate-
18 prompt us to study the class of linking games which are also potential games. Let 0 = (N ; S1; : : : ; Sn ; ) be a game in strategic form, where for each i
2 N;
Si is the strategy set of player i, and is the payo function. Let
w = (wi )i2N be a vector of positive numbers, to be called weights for the
players. A function P w : 5i2N Si ! IR is a w -potential for 0 if for every i 2 N and for all si 2 Si and ti 2 Si
i (si ; s0i ) 0 i (ti ; s0i ) = wi (P w (si ; s0i ) 0 P w (ti ; s0i ))
(11)
The game 0 is called a w-potential game if it admits a w-potential. Monderer and Shapley (1993) point out that the argmax set of a weighted potential does not depend on a particular choice of a weighted potential, and hence can be used as an equilibrium re nement. They also remark that this re nement concept is supported by some experimental results.16 Moreover, the Fictitious Play process converges to the equilibrium set in a class of games that contains the nite weighted potential games. In this section, we rst show that the class of weighted Myerson values is precisely the class of solutions which satisfy component eciency and which generate linking games that are weighted potential games. Second, we show that strategies in the argmax set of these potential games result in the formation of essentially complete cooperation structures.17 The second result, in conjunction with the results of the previous section, strengthens the case for the formation of essentially complete structures if the negotiation process is simultaneous. Some more de nitions and lemmas precede the main results of this section. 16 Monderer
and Shapley (1993) point out that this may be a mere coincidence. See also
Van Huyck et al. (1990) and Crawford (1991). 17 These results generalise analogous results of Qin (1993), who was concerned only with
Myerson values and potential games.
19 A solution is w-fair if for all cooperation structures L and for all i; j 2 N 1
wi
( i (L) 0 i (Lnffi; j gg)) =
1
wj
( j (L) 0 j (Lnffi; j gg))
(12)
partnership in (N; v) if for each T 6= S and each R N nS; v(R [ T ) = v (R): A solution18 satis es partnership consistency if for each partnership S in (N; v) and for the unanimity game uS
A coalition S is a
i (v ) = i S (v)uS for each i 2 S:
Kalai and Samet (1988) use partnership consistency as one of the conditions in their characterization of the class of weighted Shapley values. The next lemma illustrates an important property of the weighted Myerson values, and is of independent interest. We remind the reader that weighted Myerson values are weighted Shapley values of the graph-restricted game (N; vg ) (see page 5).
The weighted Myerson value w is the unique rule that is component ecient and w-fair. Lemma 3
Proof : We rst prove that w satis es the two properties mentioned. To
prove component eciency, let g = (N; L) be a cooperation structure and let S be a connected component of L. Then the associated game v g can be split up in two games vS and vN nS as follows. De ne
\ S) vN nS (T ) := vg (T nS ) vS (T ) := v g (T
for all T N . We then have vg = v S + vN nS because S is a connected component of L. Then, because all i 2 S are dummy players in the game vN nS , 18 Note
that here, we are using the term `solution' as the rule specifying payo s to the
players for classes of T U games.
20 we know by the dummy player property of the weighted Shapley value that wi (vN nS ) = 0 for all i 2 S . Similarly, wi (v S ) = 0 for all i 2 N nS . Using additivity of the weighted Shapley value we now obtain X i2S
wi (vg ) =
=
X i2S
X i2S
wi (v S ) + wi
X i2S S
wi (vN nS )
(v ) = v (N ) = v g (S ) = v (S ); S
where we use eciency of the weighted Shapley value in the third equality. To prove w-fairness, let g = (N; L) be a cooperation structure and choose ~ := Lnffi; j gg, g~ = (N; L~ ), and de ne i; j 2 N such that fi; j g 2 L. De ne L v~ := v g 0 vg~. Then, for each T v~(T ) =
N with fi; j g 6 T
X R2T ng
v(R) 0
X R2T ng~
v (R) = 0;
where we use the fact that T ng = T ng~. Hence, fi; j g is a partnership in v~. Also, since weighted Shapley values satisfy partnership consistency, we have wi (~ v ) = wi
wi (~ v ) + wj (~ v) ufi;j g
Now, note that wi
wi (~ v) + wj (~ v ) ufi;j g =
wi
wi + wj
wi (~ v) + wj (~ v)
A similar expression can be found for j . From this we see that wi (~v) wj (~v) = j wi w
This gives us ~ ) wi (~v) wj (~v) wj (L) 0 wj (L~ ) wi (L) 0 wi (L = i = j = wi w w wj We now show that there exists at most one rule that satis es component eciency and w-fairness. Suppose to the contrary that 1 and 2 are two rules
21 satisfying the two properties, and let (N; L) be a cooperation structure with a minimum number of links such that 1(L) 6= 2(L). Let fi; j g 2 L. Then by w-fairness of 1 ,
1
wi
( i1 (L) 0 i1 (Lnffi; j gg)) =
1
wj
( j1 (L) 0 j1 (Lnffi; j gg))
Hence, using the minimality of L, wj i1 (L) 0 wi j1 (L) = wj i1 (Lnffi; j gg) 0 wi j1 (Lnffi; j gg)
= wj i2 (Lnffi; j gg) 0 wi j2 (Lnffi; j gg) = wj i2 (L) 0 wi j2 (L) ; where the last equality follows from w-fairness of 2. So, now we have wj ( i1 (L) 0 i2 (L)) = wi ( j1 (L) 0 j2 (L))
(13)
It can be easily seen that equality (13) holds for all i; j that are in the same connected component of (N; L). Therefore, we can nd for each connected component S of (N; L) a number d(S ) such that for all i 2 S 1
wi
( i1 (L) 0 i2(L)) = d(S )
(14)
However, both 1 and 2 are component ecient solutions, so for each connected component S of (N; L), we have X 1 X 2
i (L) =
i (L) = v (S ) i2S
i2S
(15)
Now, combining (14) and (15), we nd that d(S ) = 0 for each connected component S , and, hence, that 1 (L) = 2(L). We now show that weighted Myerson values are the only solutions which are component ecient and which generate linking games that are weighted potential games. This follows from the previous lemma and the following lemma.
22
Let be a component ecient solution generating a linking game 0( ) that is a weighted potential game. Then there exist weights w such that is w-fair. Lemma 4
Proof : Since the linking game 0( ) is a weighted potential game, we can nd
positive weights w and a w-potential P w for the game 0( ). We will show that
is w-fair. Let L be a cooperation structure and let i; j
all k 2 N sk := fl 2 N
j fk; lg 2 Lg. Then, L(s) = L, and
2 N.
We de ne for
P w (si nfj g; sj ; s0ij ) = P w (si nfj g; sj nfig; s0ij ) = P w (si ; sj nfig; s0ij ) ;
because all these strategy tuples result in the formation of the same cooperation structure, namely Lnfi; j g. Hence,
1
wi
i (L) 0 i (Lnffi; j gg)
1
=
fi (s) 0 fi (si nfj g; s0i )
wi = P w (s) 0 P w (si nfj g; s0i )
= P w (s) 0 P w (sj nfig; s0j ) 1 = fj (s) 0 fj (sj nfig; s0j ) wj
=
1
wj
j (L) 0 j (Lnffi; j gg)
This proves that is w-fair. The following lemma shows that each weighted Myerson value generates a linking game that is a weighted potential game. Lemma 5
The linking game 0(w ) is a w-potential game.
Proof : See the appendix.
23 Combining the results we have obtained so far, we obtain our characterization of the class of weighted Myerson values. Theorem 3 Let be a component ecient solution. Then, the linking game
0( ) is a weighted potential game i is a weighted Myerson value. Remark 4: Monderer and Shapley (1993) consider
participation games,
strategic form games in which each player has the option of either cooperating with all the other players or being on his own. Such a game is in fact a linking game in which the strategy set of a player i is limited to two strategies, ; and N nfig. Monderer and Shapley prove the theorem stated above using their version of the linking game. However, the severe restriction on the players' strategy sets implies that the participation games are of limited interest. Note that in theorem 1, we proved that if satis es CE, WLS and IP, then si is a weakly dominating strategy in 0( ). Noting that weighted Myerson values satisfy these properties and any n-tuple of weakly dominating strategies must be in the argmax set of the corresponding weighted potential game, we obtain the rst part of the following theorem. Theorem 4 Let w be a set of positive weights and let P w be a weighted
potential for the linking game 0(w ). Then s 2 argmax P w . Moreover, if s 2 argmax P w , then L(s) is essentially complete for w .19
Remark 5: One can construct examples showing that the second statement
of theorem 4 cannot be strengthened. That is, if L(s) is essentially complete for w , then s is not necessarily in argmax P w . 19 The
proof of the latter half of the theorem is similar to the corresponding part of the
proof of theorem 1, and is therefore omitted.
24
Conclusion In this paper, we have studied the endogenous formation of cooperation structures in superadditive TU-games using a strategic game approach. In this strategic game, each player announces the set of players with whom he or she wants to form a link, and a link is formed if both players want to form the link. Given the resulting cooperation structure, the payo s are determined by some exogenous solution for cooperative games with cooperation structures. We have concentrated on the class of solutions satisfying three appealing properties. We have shown that in this setting both the undominated Nash equilibrium and the Coalition-Proof Nash Equilibrium predict the formation of the full cooperation structure or some payo equivalent structure. We also considered linking games that are weighted potential games and their argmax sets. It turned out that, under an eciency requirement, the class of solutions generating linking games that are weighted potential games is the class of weighted Myerson values. Further, the argmax set of the linking games that are weighted potential games predicts the formation of the full cooperation structure or some payo equivalent structure. The results obtained in this paper all point in the direction of the formation of the full cooperation structure in a superadditive environment. However, as indicated in the text, these results are sensitive to the assumptions on solutions for cooperative game with cooperation structures that we made in the paper. Further, the discussion in section 3 shows that in a context where links are formed sequentially rather than simultaneously other predictions may prevail.
25
Appendix
We provide here the proof of lemma 5. To simplify this proof, we recall some results of Kalai and Samet (1988) and we prove an intermediary claim. Kalai and Samet (1988) gave the following probabilistic de nition of the weighted Shapley value w : wi (v) =
X 26(N )
pw ( ) v(P Ri [ fig) 0 v(P Ri ) ;
where 6(N ) denotes the set of permutations of N , P Ri = fj 2 N j 01(j ) < 01 (i)g denotes the set of predecessors of player i according to , and where for each 2 6(N ) the probability pw ( ) is given by n Y
w(i) p () = Pi j =1 w (j ) i=1 w
!
With respect to the probabilities pw () we derive the following result, which plays an important role in the proof of lemma 5. De ne pw (P Ri = S [ fj g) := P w w 26(N ): P Ri =S [fj g p () and de ne p (P Rj = S [fig) correspondingly. Then Claim Let i; j
2 N and let S N nfi; j g.
pw (P Ri = S [ fj g) w = i w p (P Rj = S [ fig) wj
Proof : To simplify the proof we introduce some notation. For 2 6(N ) and
2 N we name N n(P Rk [ fkg) the tail of k in . We simply say that T is a tail in if T it the tail of k in for some k 2 N . With this notation we have
k
pw (P Ri = S [ fj g) = pw (N n (S [ fi; j g) is the tail of i in )
We denote the set N n (S [ fi; j g) by N 0Sij . Using this notation and conditional probabilities, we see that the last expression is equal to pw (N 0Sij is the tail of i j N 0Sij is a tail ) 1 pw (N 0Sij is a tail )
26 The conditional probability mentioned here is, by de nition of the probabilities pw , equal to P wi wk . Hence, we nd k2S[fi;j g
wi
pw (P Ri = S [ fj g) = P In a similar way we can show that
k 2S [fi;j g
pw (P Rj = S [ fig) = P
wj
wk
k2S [fi;j g wk
pw (N 0Sij is a tail )
(A:1)
pw (N 0Sij is a tail )
(A:2)
Combining expressions (A.1) and (A.2) gives the desired result. Lemma 5
The linking game 0(w ) is a w-potential game.
We will prove that for the game 0(w ), for i; j 2 N; s0ij and s1i ; s2i 2 Si ; s1j ; s2j 2 Sj , we have Proof :
2 5k2N nfi;jgSk
1 :=
wi (L(b)) 0 wi (L(a)) wj (L(c)) 0 wj (L(b)) + + wi wj wi (L(d)) 0 wi (L(c)) wj (L(a)) 0 wj (L(d)) + =0 (A.3) wi wj where a := (s1i ; s1j ; s0ij ); b := (s2i ; s1j ; s0ij ); c := (s2i ; s2j ; s0ij ), and d := (s1i ; s2j ; s0ij ). It can be shown analogously to theorem 2.8 of Monderer and Shapley (1993) that the property described in (A.3) is satis ed if and only if 0(w ) is a w-potential game. To prove that (A.3) is satis ed, we use the probabilistic de nition of the weighted Shapley value w of Kalai and Samet (1988). To simplify notations we will denote for each 2 fa; b; c; dg the game vL( ) by v . We consider, for the moment, the term wi (L(b)) 0 wi (L(a)) = =
X
26(N )
X
26(N )
pw () vb (P Ri [ fig) 0 v b (P Ri ) 0 va (P Ri [ fig) + va (P Ri )
pw () vb (P Ri [ fig) 0 v a(P Ri [ fig) ;
27 where the last equality follows from the fact that v a(T ) = v b (T ) for all T
N
that do not contain i. We can go through the same procedure for the other three terms appearing in 1 and this results in 1=
vb (P R [fig)0v a (P R [fig) i i
X 26(N )
pw ( )
wi
+
vd (P Ri [fig)0vc (P Ri [fig) wi
vc (P Rj [fj g)0vb (P Rj [fj g) wj
+
+
va (P Rj [fj g)0vd (P Rj [fj g) wj
Rearranging the terms, we obtain 1=
vb (P R [fig)0v c (P R [fig) i i
X 26(N )
pw ( )
wi
d a + v (P Ri [fig)w0iv (P Ri [fig) +
vc (P Rj [fj g)0v d (P Rj [fj g) wj
+
va (P Rj [fj g)0vb (P Rj [fj g) wj
Now, we can use the fact that va (S ) = v b (S ) = vc (S ) = vd (S ) for all S N nfi; j g and we obtain 1=
P
pw () 26(N ): j 2P Ri wi
P
26(N ): i2P Rj
pw () wj
v b (P Ri [ fig) 0 vc (P Ri [ fig) +
v d (P Ri [ fig) 0 va (P Ri [ fig) + v c (P Rj [ fj g) 0 v d (P Rj [ fj g) +
v a(P Rj [ fj g) 0 vb (P Rj [ fj g)
This last expression can be seen to be equal to 0 if we consider the terms per strategy pro le. Let 2 fa; b; c; dg and consider X pw ( ) pw ( ) v (P Ri [ fig) 0 v (P Rj [ fj g) 26(N ): j 2P Ri wi 26(N ): i2P Rj wj X
This is equal to X S N nfi;j g
X pw ( ) pw ( ) 0 26(N ): P Ri =S [fj g wi 26(N ): P Rj =S [fig wj
v (S [ fi; j g)
X
Now, we use the claim, and conclude that the last expression equals 0.
28
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